Finite Group Databasebeta

Subgroups — #37535565

Order 23168.

174 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153757	C₂	1	1	yes	maximal, characteristic
2	D₈×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041361	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041104	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041081	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₄₄₈	C₂²	1	1	yes	characteristic
9	D₄×Dic₁₈₁	—	2	1	no	—
10	#19153778	—	2	1	no	—
11	#19153757	C₄	1	4	yes	complemented
12	C₂×#15668424	C₂²	1	1	yes	characteristic
13	#19153750	C₂²	1	1	yes	characteristic
14	#19153737	—	2	1	no	—
15	#19153736	—	2	1	no	—
16	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₈₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
19	#107×C₁₈₁	C₂²	1	1	yes	characteristic
20	C₂×D₇₂₄	—	2	1	no	—
21	C₂²×Dic₁₈₁	—	4	1	no	—
22	C₂×C₄×D₁₈₁	—	4	1	no	—
23	D₁₄₄₈	—	2	1	no	—
24	D₁₄₄₈	C₂×C₄	1	2	yes	complemented
25	C₂×#13089678	D₄	1	1	yes	characteristic
26	C₄⋊Dic₁₈₁	—	2	1	no	—
27	C₂²⋊Dic₁₈₁	—	4	1	no	—
28	#15668426	—	4	1	no	—
29	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
30	#15668424	C₂×C₄	1	2	yes	complemented
31	C₂×D₄×C₁₈₁	—	2	1	no	—
32	D₈×C₁₈₁	—	2	1	no	—
33	D₈×C₁₈₁	C₂×C₄	1	2	yes	complemented
34	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₈₁	—	2	1	no	—
36	C₁₆×C₁₈₁	—	2	1	no	—
37	C₂²×D₁₈₁	—	4	1	no	—
38	D₇₂₄	—	2	2	no	—
39	D₇₂₄	—	4	1	no	—
40	C₂×Dic₁₈₁	—	4	1	no	—
41	C₂×Dic₁₈₁	—	8	2	no	—
42	C₂×Dic₁₈₁	D₈	1	1	yes	characteristic
43	C₂×Dic₁₈₁	D₈	1	1	yes	characteristic
44	C₄×D₁₈₁	—	8	2	no	—
45	#13089678	—	2	1	no	—
46	C₂³×C₁₈₁	—	4	1	no	—
47	D₄×C₁₈₁	—	2	2	no	—
48	D₄×C₁₈₁	—	4	1	no	—
49	C₂×C₄×C₁₈₁	—	4	1	no	—
50	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₈₁	#41	1	1	yes	characteristic
53	C₂²×C₁₈₁	—	4	2	no	—
54	C₂²×C₁₈₁	—	8	1	no	—
55	C₂²×C₁₈₁	—	8	1	no	—
56	C₂²×C₁₈₁	C₂×D₈	1	1	yes	characteristic
57	C₂×D₁₈₁	—	4	2	no	—
58	C₂×D₁₈₁	—	8	1	no	—
59	C₂×D₁₈₁	—	8	1	no	—
60	C₄×C₁₈₁	—	8	1	no	—
61	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
62	C₄×C₁₈₁	#135	1	1	yes	characteristic
63	Dic₁₈₁	—	2	1	no	—
64	Dic₁₈₁	—	8	1	no	—
65	Dic₁₈₁	D₁₆	1	2	yes	complemented
66	C₂×C₁₈₁	—	8	2	no	—
67	C₂×C₁₈₁	C₄×D₈	1	1	yes	characteristic
68	C₂×C₁₈₁	C₂×D₁₆	1	1	yes	characteristic
69	C₂×C₁₈₁	#508	1	1	yes	characteristic
70	D₁₈₁	—	8	2	no	—
71	C₁₈₁	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
72	C₄×D₁₆	—	181	1	no	maximal, Sylow
73	C₂×D₁₆	—	181	1	no	—
74	C₄×D₈	—	181	1	no	—
75	C₄×D₈	—	181	1	no	—
76	#366	—	181	1	no	—
77	#357	—	181	1	no	—
78	#357	D₁₈₁	1	1	yes	characteristic, complemented
79	C₄×C₁₆	—	181	1	no	—
80	C₂×D₈	—	181	1	no	—
81	C₂×D₈	C₂×D₁₈₁	1	1	yes	characteristic
82	C₄×D₄	—	362	1	no	—
83	C₄×D₄	—	362	1	no	—
84	D₁₆	—	181	4	no	—
85	C₂×C₁₆	—	181	1	no	—
86	C₂×C₁₆	C₂×D₁₈₁	1	1	yes	characteristic
87	#107	—	181	1	no	—
88	#107	C₂×D₁₈₁	1	1	yes	characteristic
89	#102	—	362	1	no	—
90	#102	—	362	1	no	—
91	C₄×C₈	—	181	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	362	1	no	—
94	C₂²×C₄	—	724	1	no	—
95	C₂²×C₄	—	724	1	no	—
96	D₈	—	2	1	no	—
97	D₈	—	362	1	no	—
98	D₈	—	181	2	no	—
99	D₈	C₄×D₁₈₁	1	2	yes	complemented
100	C₂×C₈	—	181	1	no	—
101	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	362	1	no	—
104	C₂²⋊C₄	—	724	1	no	—
105	C₂²⋊C₄	—	724	1	no	—
106	C₄²	—	181	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	181	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	724	1	no	—
111	D₄	—	2	2	no	—
112	D₄	—	4	1	no	—
113	D₄	—	362	2	no	—
114	D₄	—	724	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	1448	2	no	—
117	C₂×C₄	—	1448	2	no	—
118	C₂×C₄	—	181	1	no	—
119	C₂×C₄	—	181	1	no	—
120	C₂×C₄	—	724	1	no	—
121	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
122	C₈	—	362	1	no	—
123	C₈	C₂×C₄×D₁₈₁	1	1	yes	characteristic
124	C₈	#15668448	1	1	yes	characteristic
125	C₂²	—	4	2	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	—	1448	1	no	—
129	C₂²	—	1448	1	no	—
130	C₂²	—	724	2	no	—
131	C₂²	D₈×D₁₈₁	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	362	1	no	—
134	C₄	—	1448	1	no	—
135	C₄	—	181	2	no	—
136	C₄	#19153776	1	1	yes	characteristic
137	C₄	#19153837	1	1	yes	characteristic
138	C₂	—	8	2	no	—
139	C₂	—	1448	2	no	—
140	C₂	#26041359	1	1	yes	characteristic
141	C₂	D₁₆×D₁₈₁	1	1	yes	characteristic
142	C₂	#26041710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₈₁
commutator	C₈×C₁₈₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₈₁
Fitting	#357×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×D₈
prime radical	C₂×#19153757	C₂
cosocle	C₂×C₈×C₁₈₁	C₂³
hypercenter	#357	D₁₈₁
hypocenter	C₁₈₁	C₄×D₁₆
LCS	G ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. D₈×Dic₁₈₁	C₂	semidirect
3. #26041361	C₂	semidirect
4. #26041104	C₂	semidirect
5. #26041081	C₂	semidirect
6. C₁₆×Dic₁₈₁	C₂	semidirect
7. #357×C₁₈₁	C₂	semidirect
11. #19153757	C₄	semidirect
24. D₁₄₄₈	C₂×C₄	semidirect
30. #15668424	C₂×C₄	semidirect
33. D₈×C₁₈₁	C₂×C₄	semidirect
65. Dic₁₈₁	D₁₆	semidirect
71. C₁₈₁	C₄×D₁₆	semidirect
78. #357	D₁₈₁	semidirect
99. D₈	C₄×D₁₈₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19153757	2
D₈×Dic₁₈₁	2
#26041361	2
#26041104	2
#26041081	2
C₁₆×Dic₁₈₁	2
#357×C₁₈₁	2
C₄×D₁₆	181